Γ-space

In mathematics, a $$\gamma$$-space is a topological space that satisfies a certain a basic selection principle. An infinite cover of a topological space is an $$\omega$$-cover if every finite subset of this space is contained in some member of the cover, and the whole space is not a member the cover. A cover of a topological space is a $$\gamma$$-cover if every point of this space belongs to all but finitely many members of this cover. A $$\gamma$$-space is a space in which every open $$\omega$$-cover contains a $$\gamma$$-cover.

History
Gerlits and Nagy introduced the notion of γ-spaces. They listed some topological properties and enumerated them by Greek letters. The above property was the third one on this list, and therefore it is called the γ-property.

Combinatorial characterization
Let $$[\mathbb{N}]^\infty$$ be the set of all infinite subsets of the set of natural numbers. A set $$A\subset [\mathbb{N}]^\infty$$is centered if the intersection of finitely many elements of $$A$$ is infinite. Every set $$a\in[\mathbb{N}]^\infty$$we identify with its increasing enumeration, and thus the set $$a$$ we can treat as a member of the Baire space $$\mathbb{N}^\mathbb{N}$$. Therefore, $$[\mathbb{N}]^\infty$$is a topological space as a subspace of the Baire space $$\mathbb{N}^\mathbb{N}$$. A zero-dimensional separable metric space is a γ-space if and only if every continuous image of that space into the space $$[\mathbb{N}]^\infty$$that is centered has a pseudointersection.

Topological game characterization
Let $$X$$ be a topological space. The $$\gamma$$-has a pseudo intersection if there is a set game played on $$X$$ is a game with two players Alice and Bob.

1st round: Alice chooses an open $$\omega$$-cover $$\mathcal{U}_1$$ of $$X$$. Bob chooses a set $$U_1\in \mathcal{U}_1$$.

2nd round: Alice chooses an open $$\omega$$-cover $$\mathcal{U}_2$$ of $$X$$. Bob chooses a set $$U_2\in \mathcal{U}_2$$.

etc.

If $$\{U_n:n\in\mathbb{N}\}$$ is a $$\gamma$$-cover of the space $$X$$, then Bob wins the game. Otherwise, Alice wins.

A player has a winning strategy if he knows how to play in order to win the game (formally, a winning strategy is a function).

A topological space is a $$\gamma$$-space iff Alice has no winning strategy in the $$\gamma$$-game played on this space.

Properties

 * A topological space is a γ-space if and only if it satisfies $$\text{S}_1(\Omega,\Gamma)$$ selection principle.
 * Every Lindelöf space of cardinality less than the pseudointersection number $$\mathfrak{p}$$ is a $$\gamma$$-space.
 * Every $$\gamma$$-space is a Rothberger space, and thus it has strong measure zero.


 * Let $$X$$ be a Tychonoff space, and $$C(X)$$ be  the space of continuous functions $$f\colon X\to\mathbb{R}$$ with pointwise convergence topology. The space $$X$$ is a $$\gamma$$-space if and only if $$C(X)$$ is Fréchet–Urysohn  if and only if $$C(X)$$ is strong Fréchet–Urysohn.
 * Let $$A$$ be a $$\binom{\mathbf{\Omega}}{\mathbf{\Gamma}}$$ subset of the real line, and $$M$$ be a meager subset of the real line. Then the set $$A+M=\{a+x:a\in A, x\in M\}$$ is meager.